
%% bare_jrnl_compsoc.tex
%% V1.3
%% 2007/01/11
%% by Michael Shell
%% See:
%% http://www.michaelshell.org/
%% for current contact information.
%%
%% This is a skeleton file demonstrating the use of IEEEtran.cls
%% (requires IEEEtran.cls version 1.7 or later) with an IEEE Computer
%% Society journal paper.
%%
%% Support sites:
%% http://www.michaelshell.org/tex/ieeetran/
%% http://www.ctan.org/tex-archive/macros/latex/contrib/IEEEtran/
%% and
%% http://www.ieee.org/

%%*************************************************************************
%% Legal Notice:
%% This code is offered as-is without any warranty either expressed or
%% implied; without even the implied warranty of MERCHANTABILITY or
%% FITNESS FOR A PARTICULAR PURPOSE! 
%% User assumes all risk.
%% In no event shall IEEE or any contributor to this code be liable for
%% any damages or losses, including, but not limited to, incidental,
%% consequential, or any other damages, resulting from the use or misuse
%% of any information contained here.
%%
%% All comments are the opinions of their respective authors and are not
%% necessarily endorsed by the IEEE.
%%
%% This work is distributed under the LaTeX Project Public License (LPPL)
%% ( http://www.latex-project.org/ ) version 1.3, and may be freely used,
%% distributed and modified. A copy of the LPPL, version 1.3, is included
%% in the base LaTeX documentation of all distributions of LaTeX released
%% 2003/12/01 or later.
%% Retain all contribution notices and credits.
%% ** Modified files should be clearly indicated as such, including  **
%% ** renaming them and changing author support contact information. **
%%
%% File list of work: IEEEtran.cls, IEEEtran_HOWTO.pdf, bare_adv.tex,
%%                    bare_conf.tex, bare_jrnl.tex, bare_jrnl_compsoc.tex
%%*************************************************************************

% *** Authors should verify (and, if needed, correct) their LaTeX system  ***
% *** with the testflow diagnostic prior to trusting their LaTeX platform ***
% *** with production work. IEEE's font choices can trigger bugs that do  ***
% *** not appear when using other class files.                            ***
% The testflow support page is at:
% http://www.michaelshell.org/tex/testflow/




% Note that the a4paper option is mainly intended so that authors in
% countries using A4 can easily print to A4 and see how their papers will
% look in print - the typesetting of the document will not typically be
% affected with changes in paper size (but the bottom and side margins will).
% Use the testflow package mentioned above to verify correct handling of
% both paper sizes by the user's LaTeX system.
%
% Also note that the "draftcls" or "draftclsnofoot", not "draft", option
% should be used if it is desired that the figures are to be displayed in
% draft mode.
%
% The Computer Society usually requires 12pt for submissions.
%
\documentclass[10pt,journal,compsoc]{IEEEtran}
%
% If IEEEtran.cls has not been installed into the LaTeX system files,
% manually specify the path to it like:
% \documentclass[12pt,journal,compsoc]{../sty/IEEEtran}





% Some very useful LaTeX packages include:
% (uncomment the ones you want to load)


% *** MISC UTILITY PACKAGES ***
%
%\usepackage{ifpdf}
% Heiko Oberdiek's ifpdf.sty is very useful if you need conditional
% compilation based on whether the output is pdf or dvi.
% usage:
% \ifpdf
%   % pdf code
% \else
%   % dvi code
% \fi
% The latest version of ifpdf.sty can be obtained from:
% http://www.ctan.org/tex-archive/macros/latex/contrib/oberdiek/
% Also, note that IEEEtran.cls V1.7 and later provides a builtin
% \ifCLASSINFOpdf conditional that works the same way.
% When switching from latex to pdflatex and vice-versa, the compiler may
% have to be run twice to clear warning/error messages.






% *** CITATION PACKAGES ***
%
\ifCLASSOPTIONcompsoc
  % IEEE Computer Society needs nocompress option
  % requires cite.sty v4.0 or later (November 2003)
  % \usepackage[nocompress]{cite}
\else
  % normal IEEE
  % \usepackage{cite}
\fi
% cite.sty was written by Donald Arseneau
% V1.6 and later of IEEEtran pre-defines the format of the cite.sty package
% \cite{} output to follow that of IEEE. Loading the cite package will
% result in citation numbers being automatically sorted and properly
% "compressed/ranged". e.g., [1], [9], [2], [7], [5], [6] without using
% cite.sty will become [1], [2], [5]--[7], [9] using cite.sty. cite.sty's
% \cite will automatically add leading space, if needed. Use cite.sty's
% noadjust option (cite.sty V3.8 and later) if you want to turn this off.
% cite.sty is already installed on most LaTeX systems. Be sure and use
% version 4.0 (2003-05-27) and later if using hyperref.sty. cite.sty does
% not currently provide for hyperlinked citations.
% The latest version can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/cite/
% The documentation is contained in the cite.sty file itself.
%
% Note that some packages require special options to format as the Computer
% Society requires. In particular, Computer Society  papers do not use
% compressed citation ranges as is done in typical IEEE papers
% (e.g., [1]-[4]). Instead, they list every citation separately in order
% (e.g., [1], [2], [3], [4]). To get the latter we need to load the cite
% package with the nocompress option which is supported by cite.sty v4.0
% and later. Note also the use of a CLASSOPTION conditional provided by
% IEEEtran.cls V1.7 and later.



\usepackage{grffile}
 \usepackage[pdftex]{graphicx}
% *** GRAPHICS RELATED PACKAGES ***
%
\ifCLASSINFOpdf

  
  % declare the path(s) where your graphic files are
  % \graphicspath{{/graficos/}{.}}
  % and their extensions so you won't have to specify these with
  % every instance of \includegraphics
%  \DeclareGraphicsExtensions{.pdf,.jpeg,.png}
\else
  % or other class option (dvipsone, dvipdf, if not using dvips). graphicx
  % will default to the driver specified in the system graphics.cfg if no
  % driver is specified.
   %\usepackage[dvips]{graphicx}
  % declare the path(s) where your graphic files are
 %  \graphicspath{{/graficos}}
  % and their extensions so you won't have to specify these with
  % every instance of \includegraphics
  % \DeclareGraphicsExtensions{.eps}
\fi
% graphicx was written by David Carlisle and Sebastian Rahtz. It is
% required if you want graphics, photos, etc. graphicx.sty is already
% installed on most LaTeX systems. The latest version and documentation can
% be obtained at: 
% http://www.ctan.org/tex-archive/macros/latex/required/graphics/
% Another good source of documentation is "Using Imported Graphics in
% LaTeX2e" by Keith Reckdahl which can be found as epslatex.ps or
% epslatex.pdf at: http://www.ctan.org/tex-archive/info/
%
% latex, and pdflatex in dvi mode, support graphics in encapsulated
% postscript (.eps) format. pdflatex in pdf mode supports graphics
% in .pdf, .jpeg, .png and .mps (metapost) formats. Users should ensure
% that all non-photo figures use a vector format (.eps, .pdf, .mps) and
% not a bitmapped formats (.jpeg, .png). IEEE frowns on bitmapped formats
% which can result in "jaggedy"/blurry rendering of lines and letters as
% well as large increases in file sizes.
%
% You can find documentation about the pdfTeX application at:
% http://www.tug.org/applications/pdftex





% *** MATH PACKAGES ***
%
\usepackage[cmex10]{amsmath}
% A popular package from the American Mathematical Society that provides
% many useful and powerful commands for dealing with mathematics. If using
% it, be sure to load this package with the cmex10 option to ensure that
% only type 1 fonts will utilized at all point sizes. Without this option,
% it is possible that some math symbols, particularly those within
% footnotes, will be rendered in bitmap form which will result in a
% document that can not be IEEE Xplore compliant!
%
% Also, note that the amsmath package sets \interdisplaylinepenalty to 10000
% thus preventing page breaks from occurring within multiline equations. Use:
%\interdisplaylinepenalty=2500
% after loading amsmath to restore such page breaks as IEEEtran.cls normally
% does. amsmath.sty is already installed on most LaTeX systems. The latest
% version and documentation can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/required/amslatex/math/





% *** SPECIALIZED LIST PACKAGES ***
%
%\usepackage{algorithmic}
% algorithmic.sty was written by Peter Williams and Rogerio Brito.
% This package provides an algorithmic environment fo describing algorithms.
% You can use the algorithmic environment in-text or within a figure
% environment to provide for a floating algorithm. Do NOT use the algorithm
% floating environment provided by algorithm.sty (by the same authors) or
% algorithm2e.sty (by Christophe Fiorio) as IEEE does not use dedicated
% algorithm float types and packages that provide these will not provide
% correct IEEE style captions. The latest version and documentation of
% algorithmic.sty can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/algorithms/
% There is also a support site at:
% http://algorithms.berlios.de/index.html
% Also of interest may be the (relatively newer and more customizable)
% algorithmicx.sty package by Szasz Janos:
% http://www.ctan.org/tex-archive/macros/latex/contrib/algorithmicx/




% *** ALIGNMENT PACKAGES ***
%
%\usepackage{array}
% Frank Mittelbach's and David Carlisle's array.sty patches and improves
% the standard LaTeX2e array and tabular environments to provide better
% appearance and additional user controls. As the default LaTeX2e table
% generation code is lacking to the point of almost being broken with
% respect to the quality of the end results, all users are strongly
% advised to use an enhanced (at the very least that provided by array.sty)
% set of table tools. array.sty is already installed on most systems. The
% latest version and documentation can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/required/tools/


%\usepackage{mdwmath}
%\usepackage{mdwtab}
% Also highly recommended is Mark Wooding's extremely powerful MDW tools,
% especially mdwmath.sty and mdwtab.sty which are used to format equations
% and tables, respectively. The MDWtools set is already installed on most
% LaTeX systems. The lastest version and documentation is available at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/mdwtools/


% IEEEtran contains the IEEEeqnarray family of commands that can be used to
% generate multiline equations as well as matrices, tables, etc., of high
% quality.


%\usepackage{eqparbox}
% Also of notable interest is Scott Pakin's eqparbox package for creating
% (automatically sized) equal width boxes - aka "natural width parboxes".
% Available at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/eqparbox/





% *** SUBFIGURE PACKAGES ***
%\ifCLASSOPTIONcompsoc
%\usepackage[tight,normalsize,sf,SF]{subfigure}
%\else
%\usepackage[tight,footnotesize]{subfigure}
%\fi
% subfigure.sty was written by Steven Douglas Cochran. This package makes it
% easy to put subfigures in your figures. e.g., "Figure 1a and 1b". For IEEE
% work, it is a good idea to load it with the tight package option to reduce
% the amount of white space around the subfigures. Computer Society papers
% use a larger font and \sffamily font for their captions, hence the
% additional options needed under compsoc mode. subfigure.sty is already
% installed on most LaTeX systems. The latest version and documentation can
% be obtained at:
% http://www.ctan.org/tex-archive/obsolete/macros/latex/contrib/subfigure/
% subfigure.sty has been superceeded by subfig.sty.


%\ifCLASSOPTIONcompsoc
%  \usepackage[caption=false]{caption}
%  \usepackage[font=normalsize,labelfont=sf,textfont=sf]{subfig}
%\else
%  \usepackage[caption=false]{caption}
%  \usepackage[font=footnotesize]{subfig}
%\fi
% subfig.sty, also written by Steven Douglas Cochran, is the modern
% replacement for subfigure.sty. However, subfig.sty requires and
% automatically loads Axel Sommerfeldt's caption.sty which will override
% IEEEtran.cls handling of captions and this will result in nonIEEE style
% figure/table captions. To prevent this problem, be sure and preload
% caption.sty with its "caption=false" package option. This is will preserve
% IEEEtran.cls handing of captions. Version 1.3 (2005/06/28) and later 
% (recommended due to many improvements over 1.2) of subfig.sty supports
% the caption=false option directly:
%\ifCLASSOPTIONcompsoc
%  \usepackage[caption=false,font=normalsize,labelfont=sf,textfont=sf]{subfig}
%\else
%  \usepackage[caption=false,font=footnotesize]{subfig}
%\fi
%
% The latest version and documentation can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/subfig/
% The latest version and documentation of caption.sty can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/caption/




% *** FLOAT PACKAGES ***
%
%\usepackage{fixltx2e}
% fixltx2e, the successor to the earlier fix2col.sty, was written by
% Frank Mittelbach and David Carlisle. This package corrects a few problems
% in the LaTeX2e kernel, the most notable of which is that in current
% LaTeX2e releases, the ordering of single and double column floats is not
% guaranteed to be preserved. Thus, an unpatched LaTeX2e can allow a
% single column figure to be placed prior to an earlier double column
% figure. The latest version and documentation can be found at:
% http://www.ctan.org/tex-archive/macros/latex/base/



%\usepackage{stfloats}
% stfloats.sty was written by Sigitas Tolusis. This package gives LaTeX2e
% the ability to do double column floats at the bottom of the page as well
% as the top. (e.g., "\begin{figure*}[!b]" is not normally possible in
% LaTeX2e). It also provides a command:
%\fnbelowfloat
% to enable the placement of footnotes below bottom floats (the standard
% LaTeX2e kernel puts them above bottom floats). This is an invasive package
% which rewrites many portions of the LaTeX2e float routines. It may not work
% with other packages that modify the LaTeX2e float routines. The latest
% version and documentation can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/sttools/
% Documentation is contained in the stfloats.sty comments as well as in the
% presfull.pdf file. Do not use the stfloats baselinefloat ability as IEEE
% does not allow \baselineskip to stretch. Authors submitting work to the
% IEEE should note that IEEE rarely uses double column equations and
% that authors should try to avoid such use. Do not be tempted to use the
% cuted.sty or midfloat.sty packages (also by Sigitas Tolusis) as IEEE does
% not format its papers in such ways.




%\ifCLASSOPTIONcaptionsoff
%  \usepackage[nomarkers]{endfloat}
% \let\MYoriglatexcaption\caption
% \renewcommand{\caption}[2][\relax]{\MYoriglatexcaption[#2]{#2}}
%\fi
% endfloat.sty was written by James Darrell McCauley and Jeff Goldberg.
% This package may be useful when used in conjunction with IEEEtran.cls'
% captionsoff option. Some IEEE journals/societies require that submissions
% have lists of figures/tables at the end of the paper and that
% figures/tables without any captions are placed on a page by themselves at
% the end of the document. If needed, the draftcls IEEEtran class option or
% \CLASSINPUTbaselinestretch interface can be used to increase the line
% spacing as well. Be sure and use the nomarkers option of endfloat to
% prevent endfloat from "marking" where the figures would have been placed
% in the text. The two hack lines of code above are a slight modification of
% that suggested by in the endfloat docs (section 8.3.1) to ensure that
% the full captions always appear in the list of figures/tables - even if
% the user used the short optional argument of \caption[]{}.
% IEEE papers do not typically make use of \caption[]'s optional argument,
% so this should not be an issue. A similar trick can be used to disable
% captions of packages such as subfig.sty that lack options to turn off
% the subcaptions:
% For subfig.sty:
% \let\MYorigsubfloat\subfloat
% \renewcommand{\subfloat}[2][\relax]{\MYorigsubfloat[]{#2}}
% For subfigure.sty:
% \let\MYorigsubfigure\subfigure
% \renewcommand{\subfigure}[2][\relax]{\MYorigsubfigure[]{#2}}
% However, the above trick will not work if both optional arguments of
% the \subfloat/subfig command are used. Furthermore, there needs to be a
% description of each subfigure *somewhere* and endfloat does not add
% subfigure captions to its list of figures. Thus, the best approach is to
% avoid the use of subfigure captions (many IEEE journals avoid them anyway)
% and instead reference/explain all the subfigures within the main caption.
% The latest version of endfloat.sty and its documentation can obtained at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/endfloat/
%
% The IEEEtran \ifCLASSOPTIONcaptionsoff conditional can also be used
% later in the document, say, to conditionally put the References on a 
% page by themselves.




% *** PDF, URL AND HYPERLINK PACKAGES ***
%
%\usepackage{url}
% url.sty was written by Donald Arseneau. It provides better support for
% handling and breaking URLs. url.sty is already installed on most LaTeX
% systems. The latest version can be obtained at:
% http://www.ctan.org/tex-archive/macros/latex/contrib/misc/
% Read the url.sty source comments for usage information. Basically,
% \url{my_url_here}.





% *** Do not adjust lengths that control margins, column widths, etc. ***
% *** Do not use packages that alter fonts (such as pslatex).         ***
% There should be no need to do such things with IEEEtran.cls V1.6 and later.
% (Unless specifically asked to do so by the journal or conference you plan
% to submit to, of course. )


% correct bad hyphenation here
\hyphenation{op-tical net-works semi-conduc-tor u-san-do u-sa-mos}

\begin{document}
%
% paper title
% can use linebreaks \\ within to get better formatting as desired
\title{Simulacro de Combate Convencional (La batalla de Iwo Jima)}
%
%
% author names and IEEE memberships
% note positions of commas and nonbreaking spaces ( ~ ) LaTeX will not break
% a structure at a ~ so this keeps an author's name from being broken across
% two lines.
% use \thanks{} to gain access to the first footnote area
% a separate \thanks must be used for each paragraph as LaTeX2e's \thanks
% was not built to handle multiple paragraphs
%
%
%\IEEEcompsocitemizethanks is a special \thanks that produces the bulleted
% lists the Computer Society journals use for "first footnote" author
% affiliations. Use \IEEEcompsocthanksitem which works much like \item
% for each affiliation group. When not in compsoc mode,
% \IEEEcompsocitemizethanks becomes like \thanks and
% \IEEEcompsocthanksitem becomes a line break with idention. This
% facilitates dual compilation, although admittedly the differences in the
% desired content of \author between the different types of papers makes a
% one-size-fits-all approach a daunting prospect. For instance, compsoc 
% journal papers have the author affiliations above the "Manuscript
% received ..."  text while in non-compsoc journals this is reversed. Sigh.

\author{Lucila Stancato,~\IEEEmembership{I.T.B.A,}
        Damian Modernell,~\IEEEmembership{I.T.B.A,}
	Juan Brasca,~\IEEEmembership{I.T.B.A,}
        Conrado Negro,~\IEEEmembership{I.T.B.A}% <-this % stops a space
%\IEEEcompsocitemizethanks{\IEEEcompsocthanksitem M. Shell is with the Department
%of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta,
%GA, 30332.\protect\\
% note need leading \protect in front of \\ to get a newline within \thanks as
% \\ is fragile and will error, could use \hfil\break instead.
%E-mail: see http://www.michaelshell.org/contact.html
%\IEEEcompsocthanksitem J. Doe and J. Doe are with Anonymous University.}% <-this % stops a space
%\thanks{Manuscript received April 19, 2005; revised January 11, 2007.}
}

% note the % following the last \IEEEmembership and also \thanks - 
% these prevent an unwanted space from occurring between the last author name
% and the end of the author line. i.e., if you had this:
% 
% \author{....lastname \thanks{...} \thanks{...} }
%                     ^------------^------------^----Do not want these spaces!
%
% a space would be appended to the last name and could cause every name on that
% line to be shifted left slightly. This is one of those "LaTeX things". For
% instance, "\textbf{A} \textbf{B}" will typeset as "A B" not "AB". To get
% "AB" then you have to do: "\textbf{A}\textbf{B}"
% \thanks is no different in this regard, so shield the last } of each \thanks
% that ends a line with a % and do not let a space in before the next \thanks.
% Spaces after \IEEEmembership other than the last one are OK (and needed) as
% you are supposed to have spaces between the names. For what it is worth,
% this is a minor point as most people would not even notice if the said evil
% space somehow managed to creep in.



% The paper headers
%\markboth{Journal of \LaTeX\ Class Files,~Vol.~6, No.~1, January~2007}%
%{Shell \MakeLowercase{\textit{et al.}}: Bare Demo of IEEEtran.cls for Computer Society Journals}
% The only time the second header will appear is for the odd numbered pages
% after the title page when using the twoside option.
% 
% *** Note that you probably will NOT want to include the author's ***
% *** name in the headers of peer review papers.                   ***
% You can use \ifCLASSOPTIONpeerreview for conditional compilation here if
% you desire.



% The publisher's ID mark at the bottom of the page is less important with
% Computer Society journal papers as those publications place the marks
% outside of the main text columns and, therefore, unlike regular IEEE
% journals, the available text space is not reduced by their presence.
% If you want to put a publisher's ID mark on the page you can do it like
% this:
%\IEEEpubid{0000--0000/00\$00.00~\copyright~2007 IEEE}
% or like this to get the Computer Society new two part style.
%\IEEEpubid{\makebox[\columnwidth]{\hfill 0000--0000/00/\$00.00~\copyright~2007 IEEE}%
%\hspace{\columnsep}\makebox[\columnwidth]{Published by the IEEE Computer Society\hfill}}
% Remember, if you use this you must call \IEEEpubidadjcol in the second
% column for its text to clear the IEEEpubid mark (Computer Society jorunal
% papers don't need this extra clearance.)



% use for special paper notices
%\IEEEspecialpapernotice{(Invited Paper)}



% for Computer Society papers, we must declare the abstract and index terms
% PRIOR to the title within the \IEEEcompsoctitleabstractindextext IEEEtran
% command as these need to go into the title area created by \maketitle.
\IEEEcompsoctitleabstractindextext{%
\begin{abstract}
%\boldmath
  Bas\'andonos en la ley cuadr\'atica de Lanchester modelamos la Batalla de Iwo Jima durante la Segunda Guerra Mundial.
 Mostramos que el modelo enunciado por Lanchester se aproxima a los datos hist\'oricos. 
 Tambi\'en usamos el modelo para simular otras posibilidades de ataque por parte de Estados Unidos.
\end{abstract}
% IEEEtran.cls defaults to using nonbold math in the Abstract.
% This preserves the distinction between vectors and scalars. However,
% if the journal you are submitting to favors bold math in the abstract,
% then you can use LaTeX's standard command \boldmath at the very start
% of the abstract to achieve this. Many IEEE journals frown on math
% in the abstract anyway. In particular, the Computer Society does
% not want either math or citations to appear in the abstract.

% Note that keywords are not normally used for peerreview papers.
\begin{IEEEkeywords}
Simulaci\'on, Modelo de Lanchester, Combate convencional.
\end{IEEEkeywords}}


% make the title area
\maketitle


% To allow for easy dual compilation without having to reenter the
% abstract/keywords data, the \IEEEcompsoctitleabstractindextext text will
% not be used in maketitle, but will appear (i.e., to be "transported")
% here as \IEEEdisplaynotcompsoctitleabstractindextext when compsoc mode
% is not selected <OR> if conference mode is selected - because compsoc
% conference papers position the abstract like regular (non-compsoc)
% papers do!
\IEEEdisplaynotcompsoctitleabstractindextext
% \IEEEdisplaynotcompsoctitleabstractindextext has no effect when using
% compsoc under a non-conference mode.


% For peer review papers, you can put extra information on the cover
% page as needed:
% \ifCLASSOPTIONpeerreview
% \begin{center} \bfseries EDICS Category: 3-BBND \end{center}
% \fi
%
% For peerreview papers, this IEEEtran command inserts a page break and
% creates the second title. It will be ignored for other modes.
\IEEEpeerreviewmaketitle



\section{Introducci\'on}
% Computer Society journal papers do something a tad strange with the very
% first section heading (almost always called "Introduction"). They place it
% ABOVE the main text! IEEEtran.cls currently does not do this for you.
% However, You can achieve this effect by making LaTeX jump through some
% hoops via something like:
%
%\ifCLASSOPTIONcompsoc
%  \noindent\raisebox{2\baselineskip}[0pt][0pt]%
%  {\parbox{\columnwidth}{\section{Introduction}\label{sec:introduction}%
%  \global\everypar=\everypar}}%
%  \vspace{-1\baselineskip}\vspace{-\parskip}\par
%\else
%  \section{Introduction}\label{sec:introduction}\par
%\fi
%
% Admittedly, this is a hack and may well be fragile, but seems to do the
% trick for me. Note the need to keep any \label that may be used right
% after \section in the above as the hack puts \section within a raised box.



% The very first letter is a 2 line initial drop letter followed
% by the rest of the first word in caps (small caps for compsoc).
% 
% form to use if the first word consists of a single letter:
% \IEEEPARstart{A}{demo} file is ....
% 
% form to use if you need the single drop letter followed by
% normal text (unknown if ever used by IEEE):
% \IEEEPARstart{A}{}demo file is ....
% 
% Some journals put the first two words in caps:
% \IEEEPARstart{T}{his demo} file is ....
% 
% Here we have the typical use of a "T" for an initial drop letter
% and "HIS" in caps to complete the first word.
\IEEEPARstart{D}{}urante la primera guerra mundial, Frederick W. Lanchester se interes\'o en predecir el
 resultado de ataques aereos.  Para hacerlo  describi\'o una serie de ecuaciones diferenciales, que hoy se
 conocen como \emph{Leyes de Poder de Lanchester}�. 
  Estas leyes describen como se atacar\'ian estas fuerzas.
  Pero las armas modernas cambiaron el modelo que inicialmente era lineal, por 
 el modelo cuadr\'atico, dado que las distancias de ataque y la cantidad de heridos posibles cambian.\\
\indent En la secci\'on 2 mostramos el modelo cuadr\'atico de Lanchester, que modela la cantidad de tropas en 
funci\'on del tiempo. 
 Aplicamos este modelo para simular la Batalla de Iwo Jima durante la Segunda Guerra Mundial.
Usando el mismo modelo, analizamos dos estrategias de ataque alternativas en la secci\'on 3, 
 sin cambiar los par\'ametros del sistema, pero cambiando el r\'egimen de refuerzos de Estados Unidos.
 De esta manera determinamos que la pol\'itica de refuerzos influye en la estrategia de combate de una
batalla convencional y tambi\'en la importancia de la cantidad de tropas por sobre el coeficiente de 
efectividad de las tropas.
% You must have at least 2 lines in the paragraph with the drop letter
% (should never be an issue)

\section{Modelo cuadr\'atico de Lanchester}
El modelo de simulaci\'on que proponemos para un combate convencional esta dado por las ecuaciones diferenciales
 {\ref{eq:dif3}} y  {\ref{eq:dif4}}. $\alpha$ es el coeficiente de efectividad en combate de la fuerza $y$, 
y $\beta$ el de la fuerza $x$. Las constantes $a$ y $b$ representan las tasas unitarias de p\'erdidas operativas y $f(t)$ y 
$g(t)$ representan los refuerzos de ambas fuerzas.
\begin{align}
\label{eq:dif3}
\dot{x} &= -ax -\alpha y + f(t) \\
\label{eq:dif4}
\dot{y} &= -by -\beta x + g(t)
\end{align}
Si no tenemos en cuenta las p\'erdidas operativas  ni los refuerzos en ninguno de los ej\'ercitos, obtenemos las 
ecuaciones {\ref{eq:xpunto}} y {\ref{eq:ypunto}}, que es el modelo cuadr\'atico propuesto por Lanchester.
\begin{equation}
\label{eq:xpunto}
 \frac{dx}{dt} = -\alpha y
\end{equation}
\begin{equation}
\label{eq:ypunto}
 \frac{dy}{dt} = -\beta x
\end{equation}


%Utilizando este modelo, podemos demostrar que:
%\begin{equation}
%\label{eq:demofin}
%\alpha x^2 - \beta y^2 = K
%\end{equation}
%donde K es constante.\\
Partiendo de las ecuaciones {\ref{eq:xpunto}} y {\ref{eq:ypunto}}
dividimos miembro a miembro para obtener la ecuaci\'on {\ref{eq:demopaso1}}
\begin{equation}
\label{eq:demopaso1}
\frac{\frac{dx}{dt}}{\frac{dy}{dt}} = \frac{-\beta y}{-\alpha x}
\end{equation}
De {\ref{eq:demopaso1}} deducimos la igualdad de la ecuaci\'on {\ref{eq:demopaso2}}
\begin{equation}
\label{eq:demopaso2}
-\alpha x dx = -\beta y dy
\end{equation}
Integrando miembro a miembro obtenemos la expresi\'on en {\ref{eq:demopaso3}}
\begin{equation}
\label{eq:demopaso3}
\alpha x^2 - \beta y^2 = K
\end{equation}

K representa una \emph{constante de fuerza de batalla} que permanece
invariante a lo largo del combate. Por lo tanto, podemos ver
que:

\begin{enumerate}
 \item  El resultado del combate esta dado \'unicamente por la efectividad y la cantidad de tropas de cada bando.
 \item  La cantidad de tropas es mucho m\'as importante que su efectividad.
\end{enumerate}


\begin{figure}[t]
\label{fig:grafico_k}
%\begin{center}
\centering
\includegraphics[width=3.2in]{grafico_k}
% where an .eps filename suffix will be assumed under latex, 
% and a .pdf suffix will be assumed for pdflatex; or what has been declared
% via \DeclareGraphicsExtensions.
\caption{Gr\'afica cualitativa de combate de dos fuerzas seg\'un el valor de K}
%\end{center}
\end{figure}


En la figura 1 mostramos como 2 ej\'ercitos comienzan el combate con la misma cantidad 
de unidades.  Cambiando el valor de la constante \emph{K} obtenemos las distintas curvas. Observamos que
si $K>0$ el ej\'ercito $x$ gana la batalla, en cambio, si $K<0$, el ej\'ercito que gana la batalla es el $y$.
En el caso en que $K=0$, las fuerzas se destruyen mutuamente, y la batalla resulta en un empate.  Cabe aclarar que 
el modelo no tiene en cuenta otros factores, que pueden llevar a que un ej\'ercito se rinda, o a un 
pacto de no agresi\'on, o a la posibilidad de aliarse con otro ej\'ercito.

\section{Simulaci\'on de combate en la isla de Iwo Jima}
 
La batalla de Iwo Jima tuvo lugar desde el 16 de febrero de 1945 hasta el 26 de marzo del mismo a\~no.
 Fue la batalla m\'as decisiva de la Guerra del Pac\'ifico en la Segunda Guerra Mundial, entre Estados Unidos y 
el Imperio del Jap\'on.\\
\indent La isla de Iwo Jima se encuentra 1200 km al sur de Tokio, y fue de suma importancia para los Estados Unidos 
la conquista de la misma debido a su ubicaci\'on geogr\'afica, ya que les permiti\'o a los bombarderos B-29 
despegar desde la isla y tener autonom\'ia de vuelo para poder bombardear Jap\'on. Por otro lado, para los japoneses,
 la isla significaba una base a\'erea en el pac\'ifico y radares que les permit\'ian identificar cualquier
 avi\'on o nave enemiga que quisiera aproximarse a Jap\'on.\\
\indent En el transcurso de la batalla, las fuerzas norteamericanas tuvieron varias oleadas de refuerzos, llegando a
 un n\'umero total de 73000 tropas, mientras que los japoneses, 
no ten\'ian manera de reforzarse ya que la mayor\'ia de sus barcos de transporte habian sido hundidos por submarinos
 aliados.\\
\indent Como mencionamos en la secci\'on 2, el modelo de sistema que empleamos para simular la batalla corresponde a un modelo de combate convencional
 donde $x(t)$ e $y(t)$ representan
las magnitudes de las fuerzas Norteamericanas y Japonesas respectivamente. $\alpha $ y $\beta$ corresponden a 
los coeficientes de efectividad en combate de la fuerza enemiga. $f(t)$ modela la pol\'itica de refuerzos del
 ej\'ercito de los Estados Unidos. Jap\'on no tiene refuerzos. Se desprecian las p\'erdidas operativas de ambos.\\
El modelo lo expresamos en las ecuaciones diferenciales \ref{eq:dif1} y \ref{eq:dif2}
\begin{align}
\label{eq:dif1}
\dot{x} &= -\alpha y + f(t) \\
\label{eq:dif2}
\dot{y} &= -\beta x
\end{align}

\indent Al inicio de la batalla, Jap\'on contaba con una fuerza de aproximadamente 21500 tropas que sab\'ian de antemano que no 
regresar\'ian con vida a territorio patrio, y cuyas \'ordenes eran pelear hasta el \'ultimo hombre. 
Por este motivo las fuerzas niponas tienen un coeficiente de efectividad mayor
al de las fuerzas aliadas, que lo determinamos como $ \alpha = 0.0544 dia^{-1}$. Estados Unidos desembarc\'o en
 la isla con 54000 hombres y tuvo refuerzos  que son
representados por la funci\'on $f(t)$ en \ref{eq:f(t)}. Su coeficiente de efectividad lo determinamos 
en $\beta = 0.0106dia^{-1}$.

\begin{equation}
\label{eq:f(t)}
f(t) = \left\{
 \begin{array}{rl}
  54000 \hspace{0.5cm} 0 < t < 1\\
   0 \hspace{0.5cm} 1 < t < 2\\
   6000 \hspace{0.5cm} 2 < t < 3\\
  0 \hspace{0.5cm} 3 < t < 5\\
13000 \hspace{0.5cm} 5 < t < 6\\
 0 \hspace{1.15cm} t \geq 6\\
 \end{array} \right.
\end{equation}
% needed in second column of first page if using \IEEEpubid
%\IEEEpubidadjcol

\indent Realizamos la simulaci\'on num\'erica del modelo y obtenemos los resultados que ilustramos
en la figura 2. Tambi\'en incluimos en la figura 2 los valores hist\'oricos de la magnitud de las fuerzas Norteamericanas
para distintos d\'ias durante la batalla. Mostramos estos datos en la tabla \ref{tab:sim}.

% An example of a floating table. Note that, for IEEE style tables, the 
% \caption command should come BEFORE the table. Table text will default to
% \footnotesize as IEEE normally uses this smaller font for tables.
% The \label must come after \caption as always.
%
\begin{table}[!t]
%% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
\caption{Fuerzas Norteamericanas}
\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{c c}
\hline
\hline
Tiempo ( d\'ias ) & Fuerza USA\\
\hline
7 &  66000\\
9 & 62000 \\
12 & 58500 \\
19 & 56200  \\
24 & 54000 \\
\hline
\hline
\end{tabular}
\label{tab:sim}
\end{table}


\begin{figure}[t]
\label{fig:sim1}
%\begin{center}
\centering
\includegraphics[width=3.2in]{simulacro}
% where an .eps filename suffix will be assumed under latex, 
% and a .pdf suffix will be assumed for pdflatex; or what has been declared
% via \DeclareGraphicsExtensions.
\caption{Simulaci\'on de la batalla de Iwo Jima}
%\end{center}
\end{figure}

\section{Pol\'iticas de refuerzos}
\indent Estados Unidos envi\'o 73000 hombres a la batalla de Iwo Jima.  La estrategia de refuerzos que emple\'o tuvo un saldo de 53193 hombres
que regresaron con vida.
Usando el modelo cuadr\'atico de Lanchester, simulamos dos pol\'iticas de refuerzos distintas.  El modelo que usamos es el mismo, lo que cambia
es la cantidad y el momento en el que Estados Unidos env\'ia refuerzos a la isla de Iwo Jima.
\subsection{Todos los soldados juntos}
\indent La primer pol\'itica que simulamos consiste en mandar 73000 soldados norteamericanos el primer d\'ia de combate.
La simulaci\'on se representa en la figura 3 y muestra que la batalla tiene la misma duraci\'on que la batalla real. 
La diferencia se encuentra en que, al final de la batalla,
 las tropas estadounidenses alcanzan los 54377 hombres, 1184 hombres m\'as que con la estrategia original.  

Enviar todas las tropas el mismo d\'ia implica tener recursos suficientes (aviones, municiones, etc.).  Adem\'as, s\'olo se podr\'ia predecir
el resultado de un ataque una vez que se conocen los par\'ametros del sistema $\alpha$ y $\beta$.  Como esos par\'ametros pueden estimarse
una vez realizada la batalla, enviar 73000 hombres en el primer momento podr\'ia parecer excesivo.
 
\begin{figure}[t]
\label{fig:sim1}
%\begin{center}
\centering
\includegraphics[width=3.2in]{ref1}
% where an .eps filename suffix will be assumed under latex, 
% and a .pdf suffix will be assumed for pdflatex; or what has been declared
% via \DeclareGraphicsExtensions.
\caption{Primer pol\'itica de refuerzos}
%\end{center}
\end{figure}
   
\subsection{Refuerzos tard\'ios}
La segunda pol\'itica de refuerzos que simulamos consiste en enviar las tropas en cuatro momentos distintos. 
 En la tabla 2 mostramos la cantidad de hombres enviados a la isla de Iwo Jima con esta estrategia.  

\begin{table}[!t]
%% increase table row spacing, adjust to taste
\renewcommand{\arraystretch}{1.3}
% if using array.sty, it might be a good idea to tweak the value of
% \extrarowheight as needed to properly center the text within the cells
\caption{Segunda Pol\'itiaca de Refuerzos Norteamericanos}
\centering
%% Some packages, such as MDW tools, offer better commands for making tables
%% than the plain LaTeX2e tabular which is used here.
\begin{tabular}{c c}
\hline
\hline
Tiempo [ d\'ias ] & Tropas enviados\\
\hline
1 &  21500\\
9 &  20000 \\
13 & 10000 \\
21 & 10000  \\
\hline
\hline
\end{tabular}
\label{tab:refuerzos2}
\end{table}


En la figura 4 observamos que la duraci\'on de la batalla es aproximadamente el doble a la anterior. Adem\'as, 
la cantidad de bajas en este escenario es aproximadamente de 51000 soldados, contra los aproximadamente 20000 que mueren en la simulaci\'on original.  Esto implica que, no s\'olo se pierden 
m\'as vidas, sino que se gasta m\'as, porque se necesitan m\'as municiones, m\'as comida y recursos para abastecer a los soldados durante 65 d\'ias en lugar de 
los 35 que dur\'o el combate.

 
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\begin{figure}[t]
\label{fig:refuerzos2}
%\begin{center}
\centering
\includegraphics[width=3.2in]{grafico3_bis}
% where an .eps filename suffix will be assumed under latex, 
% and a .pdf suffix will be assumed for pdflatex; or what has been declared
% via \DeclareGraphicsExtensions.
\caption{Segunda politica de refuerzos}
%\end{center}
\end{figure}


% Note that IEEE typically puts floats only at the top, even when this
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% An example of a double column floating figure using two subfigures.
% (The subfig.sty package must be loaded for this to work.)
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% \label for the overall figure must come after \caption.
% \hfil must be used as a separator to get equal spacing.
% The subfigure.sty package works much the same way, except \subfigure is
% used instead of \subfloat.
%
%\begin{figure*}[!t]
%\centerline{\subfloat[Case I]\includegraphics[width=2.5in]{subfigcase1}%
%\label{fig_first_case}}
%\hfil
%\subfloat[Case II]{\includegraphics[width=2.5in]{subfigcase2}%
%\label{fig_second_case}}}
%\caption{Simulation results}
%\label{fig_sim}
%\end{figure*}
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\section{Conclusi\'on}
%Con el modelo que propusimos de la batalla de Iwo Jima, expresado por las ecuaciones diferenciales 8 y 9, 
%nos aproximamos considerablemente a los valores reales. Sin embargo, el modelo puede ser mejorado agregando a las ecuaciones factores que 
%pueden influir en la cantidad de bajas de cada ej\'ercito, como por ejemplo estimar una tasa de p\'erdidas operativas
%que puede depender de la calidad m\'edica, la alimentaci\'on, o las deserciones de ambas fuerzas.

El modelo que usamos para simular la Batalla de Iwo Jima se basa en las leyes de poder de Lanchester. La 
simulaci\'on de la batalla utilizando este modelo se aproxima considerablemente a los datos hist\'oricos. Sin embargo, el 
modelo puede ser mejorado agregando a las ecuaciones factores que 
pueden influir en la cantidad de bajas de cada ej\'ercito, como por ejemplo estimar una tasa de p\'erdidas operativas
que puede depender de la calidad m\'edica, la alimentaci\'on, o las deserciones de ambas fuerzas. Comprobamos que aunque el ej\'ercito
Japon\'es tuviera un coeficiente de combate mayor al de Estados Unidos, pierde la batalla porque Estados Unidos tiene m\'as tropas.\\
\indent Al ensayar distintas pol\'iticas de refuerzos, podemos ver que si se env\'ian todos los hombres al combate en
el mismo momento, la batalla dura menos que cuando los refuerzos tardan en llegar. Esto responde a que la ley 
de Lanchester es cuadr\'aticamente proporcional a la cantidad de tropas. Es por esto, que la segunda pol\'itica
de refuerzos muestra una batalla m\'as duradera. No solo eso, sino que podemos ver que al inicio de la batalla,
el ejercito Japon\'es supera a los Estados Unidos. Esto es una evidencia de que el coeficiente de combate Japon\'es
es mayor al de los Estados Unidos. Y es por eso, que el ej\'ercito Estadounidense necesita enviar m\'as tropas.






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\bibitem{IEEEhowto:kopka}
 Lanchester, Frederick William. \emph{Aircraft in Warfare: The Dawn of the Fourth Arm.}


\end{thebibliography}

\end{document}


